Near-optimal Lower Bounds on Quantifier Depth and Weisfeiler-Leman Refinement Steps
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We prove near-optimal tradeoffs for quantifier depth (also called quantifier rank) versus number of variables in first-order logic by exhibiting pairs of n-element structures that can be distinguished by a k-variable first-order sentence but where every such sentence requires quantifier depth at least nω (k/log k). Our tradeoffs also apply to first-order counting logic and, by the known connection to the k-dimensional Weisfeiler-Leman algorithm, imply near-optimal lower bounds on the number of refinement iterations.A key component in our proof is the hardness condensation technique introduced by Razborov in the context of proof complexity. We apply this method to reduce the domain size of relational structures while maintaining the minimal quantifier depth needed to distinguish them in finite variable logics.
Original language | English |
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Article number | 32 |
Journal | Journal of the ACM |
Volume | 70 |
Issue number | 5 |
Pages (from-to) | 1-32 |
ISSN | 0004-5411 |
DOIs | |
Publication status | Published - 2023 |
Bibliographical note
Publisher Copyright:
© 2023 Association for Computing Machinery.
- bounded variable fragment, first-order counting logic, First-order logic, hardness condensation, lower bounds, quantifier depth, refinement iterations, tradeoffs, Weisfeiler-Leman, XORification
Research areas
ID: 372610177